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Classes of Submodular Constraints Expressible by Graph Cuts

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Principles and Practice of Constraint Programming (CP 2008)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 5202))

Abstract

Submodular constraints play an important role both in theory and practice of valued constraint satisfaction problems (VCSPs). It has previously been shown, using results from the theory of combinatorial optimisation, that instances of VCSPs with submodular constraints can be minimised in polynomial time. However, the general algorithm is of order O(n 6) and hence rather impractical. In this paper, by using results from the theory of pseudo-Boolean optimisation, we identify several broad classes of submodular constraints over a Boolean domain which are expressible using binary submodular constraints, and hence can be minimised in cubic time. We also discuss the question of whether all submodular constraints of bounded arity over a Boolean domain are expressible using only binary submodular constraints, and can therefore be minimised efficiently.

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References

  1. Billionet, A., Minoux, M.: Maximizing a supermodular pseudo-boolean function: a polynomial algorithm for cubic functions. D. App. Mathematics 12, 1–11 (1985)

    Article  Google Scholar 

  2. Bistarelli, S., Fargier, H., Montanari, U., Rossi, F., Schiex, T., Verfaillie, G.: Semiring-based CSPs and valued CSPs: Frameworks, properties, and comparison. Constraints 4, 199–240 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boros, E., Hammer, P.L.: Pseudo-boolean optimization. Discrete Applied Mathematics 123(1-3), 155–225 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bulatov, A., Krokhin, A., Jeavons, P.: Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing 34(3), 720–742 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cohen, D., Cooper, M., Jeavons, P.: An algebraic characterisation of complexity for valued constraints. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 107–121. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  6. Cohen, D., Cooper, M., Jeavons, P.: Generalising submodularity and Horn clauses: Tractable optimization problems defined by tournament pair multimorphisms. Theoretical Computer Science (in press, 2008)

    Google Scholar 

  7. Cohen, D., Cooper, M., Jeavons, P., Krokhin, A.: Supermodular functions and the complexity of Max-CSP. Discrete Applied Mathematics 149, 53–72 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cohen, D., Cooper, M., Jeavons, P., Krokhin, A.: The complexity of soft constraint satisfaction. Artificial Intelligence 170, 983–1016 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cooper, M.C.: Minimization of locally defined submodular functions by optimal soft arc consistency. Constraints 13 (2008)

    Google Scholar 

  10. Cooper, M.: High-order consistency in valued constraint satisfaction. Constraints 10, 283–305 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Creignou, N., Khanna, S., Sudan, M.: Complexity Classification of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications, vol. 7. SIAM, Philadelphia (2001)

    Google Scholar 

  12. Gallo, G., Simeone, B.: On the supermodular knapsack problem. Mathematical Programming 45, 295–309 (1988)

    Article  MathSciNet  Google Scholar 

  13. Goldberg, A., Tarjan, R.: A new approach to the maximum flow problem. Journal of the ACM 35, 921–940 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Iwata, S.: Submodular function minimization. Math. Progr. 112, 45–64 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jeavons, P., Cohen, D., Cooper, M.: Constraints, consistency and closure. Artificial Intelligence 101(1–2), 251–265 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts? IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 147–159 (2004)

    Article  Google Scholar 

  17. Montanari, U.: Networks of constraints: Fundamental properties and applications to picture processing. Information Sciences 7, 95–132 (1974)

    Article  MathSciNet  Google Scholar 

  18. Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization (1988)

    Google Scholar 

  19. Orlin, J.B.: A faster strongly polynomial time algorithm for submodular function minimization. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 240–251. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  20. Rhys, J.: A selection problem of shared fixed costs and network flows. Management Science 17(3), 200–207 (1970)

    Article  MATH  Google Scholar 

  21. Rossi, F., van Beek, P., Walsh, T. (eds.): The Handbook of CP. Elsevier, Amsterdam (2006)

    Google Scholar 

  22. Schiex, T., Fargier, H., Verfaillie, G.: Valued constraint satisfaction problems: hard and easy problems. In: IJCAI 1995, pp. 631–639 (1995)

    Google Scholar 

  23. Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. of Combinatorial Theory, Series B 80, 346–355 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  25. Zalesky, B.: Efficient determination of Gibbs estimators with submodular energy functions. arXiv:math/0304041v1 (February 2008)

    Google Scholar 

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Peter J. Stuckey

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Živný, S., Jeavons, P.G. (2008). Classes of Submodular Constraints Expressible by Graph Cuts. In: Stuckey, P.J. (eds) Principles and Practice of Constraint Programming. CP 2008. Lecture Notes in Computer Science, vol 5202. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85958-1_8

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  • DOI: https://doi.org/10.1007/978-3-540-85958-1_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85957-4

  • Online ISBN: 978-3-540-85958-1

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